A curve that has torsion 
A curve that has torsion zero everywhere lies in this many dimensions. The Hilbert and Peano curves have a Hausdorff dimension of this. A trivial statement about spheres of this many dimensions is also true for the dimension one greater than this one by the Poincaré conjecture. An Argand diagram requires this many dimensions, since the (*) complex numbers are a vector space of this many dimensions over the reals. A manifold of this many dimensions is called a surface. Polygons and polar coordinates exist in this many dimensions, and objects of this many dimensions have a volume of zero, as they have no depth. For 10 points, give the number of dimensions of a plane.
A curve that has torsion zero everywhere lies in this many dimensions.
{two} dimensions